## Derivation of the spherical polar Laplacian

Posted by peeterjoot on October 9, 2010

[Click here for a PDF of this post with nicer formatting]

# Motivation.

In [1] was a Geometric Algebra derivation of the 2D polar Laplacian by squaring the quadient. In [2] was a factorization of the spherical polar unit vectors in a tidy compact form. Here both these ideas are utilized to derive the spherical polar form for the Laplacian, an operation that is strictly algebraic (squaring the gradient) provided we operate on the unit vectors correctly.

# Our rotation multivector.

Our starting point is a pair of rotations. We rotate first in the plane by

Then apply a rotation in the plane

The composition of rotations now gives us

# Expressions for the unit vectors.

The unit vectors in the rotated frame can now be calculated. With we can calculate

Performing these we get

and

and

Summarizing these are

# Derivatives of the unit vectors.

We’ll need the partials. Most of these can be computed from 3.9 by inspection, and are

# Expanding the Laplacian.

We note that the line element is , so our gradient in spherical coordinates is

We can now evaluate the Laplacian

Evaluating these one set at a time we have

and

and

Summing these we have

This is often written with a chain rule trick to considate the and partials

It’s simple to verify that this is identical to 5.23.

# References

[1] Peeter Joot. Polar form for the gradient and Laplacian. [online]. http://sites.google.com/site/peeterjoot/math2009/polarGradAndLaplacian.pdf.

[2] Peeter Joot. Spherical Polar unit vectors in exponential form. [online]. http://sites.google.com/site/peeterjoot/math2009/sphericalPolarUnit.pdf .

## Leave a Reply